I have these $n$ equations, with $n$ variables. Variables are first $n$ positive integers, constants can be any rational number including zero. Given that there is always a solution, how do we find a ...

I am new to deep learning. I am reading the book by Ian Goodfellow called "Deep Learning". I am in chapter 1 about Linear Algebra. There in the section called Linear Dependence and Span they say that ...

In Section 7.6.2 of the textbook "Algorithms" by Sanjoy Dasgupta, Christos Papadimitriou, and Umesh Vazirani, the authors provide a geometric interpretation of the two main tasks of each iteration of ...

I know the best algorithm to solve a linear system in $\mathbb{R}$ with $n$ variables is Coppersmith-Winograd's algorithm, which has a complexity of
$$
O\left(n^{2.376}\right).
$$
How much easier is ...

If I have some collection of bits, -- a byte, say -- of arbitrary value then I can transform it into some other value by means of exclusive-oring it with a subset of (in this case) eight fixed values, ...

I have an n by n symmetric matrix, and I would like to compute its square in as small a circuit complexity as possible. It's sparse: there are sqrt(n) nonzero entries in each row/column, so the input ...

This is a homework question, so I'm not looking for answers, just general guidance.
I'm looking at a Sublinear Algorithms survey where (Group) Homomorphism property testing is discussed. The case of ...

Is there is an efficient algorithm to solve the following optimization:
$\mathbf{x}^* = \arg\min_\mathbf{x}\sum_i ||\mathbf{A_i x} - \mathbf{b_i}||_1$
for given $\mathbf{b_i}, \mathbf{A_i}\ \forall ...

I'm trying to solve an integer linear program (ILP) in which a constraint of the following kind must be met:
$x_1 \oplus x_2 \oplus \cdots \oplus x_n = 1$
where $\oplus$ is the binary xor operator.
...

I'm going through a tutorial that is using the Game of Life as example code. It has a function in it that finds the neighbor of a given cell. It is explained quickly that "When applying a delta of -1, ...

First Let's take a look at the convolution $\displaystyle C _ { i } = \sum _ { j \oplus k = i } A _ { j } * B _ { k }$, and the $\oplus$represents any boolean operation. And we are able to evaluate $C$...

Given
I have a $n$ dimensional $\vec{a}$.
All elements of $\vec{a}$ are between 0 and a positive number $K$.
$n$ is about 15 to 20.
Problem
I want to randomly and unbiasedly choose a vector $\vec{b}...

Shapes
Let $C$ be the unit hypercube in $\mathbb{R}^{n}$.
Let $\vec{o}$ be a point in $\mathbb{R}^{n}$.
Let $B$ be a $n \times m$ matrix. The columns of $B$ are a set of linearly independent vectors ...

Suppose the set of linear inequalities $Ax\le b$, in which $A\in\mathbb{R}^{m\times n},x,b\in\mathbb{R}^n$ is given. Is it possible to determine in polynomial time with regard to $m$ and $n$ if there ...

In Chapter 3 of Mining of Massive Datasets, the basis of locality sensitive hashing is explained. They notably mention simhash for the cosine distance, where random hyperplanes are generated, and for ...

I'm trying to implement this algorithm but I'm having problems reproducing the exemple that it gives a solution to.
The general method that I tried is:
Make a grid $\theta \in [0,2\pi)$, with say N=...

I'm looking for an algorithm that can find any matrix $a_{j,i}$ such that
$$
\sum_{i \in I} \left(\sum_{j\in J} a_{j,i}\right)^2
$$
is minimal, while also for each $j\in J$ satisfying the constraint
...

I want to know what are the efficient way to invert a Sparse Matrix? Are there any algorithm,linear algebra or expansions that make this task easier with out actually inverting the matrix?
Thank you ...

Assume also that $v_1,v_2$ are linearly independent, and $q_i \in \mathbb{R}^n$ denotes the $i$-th column of $Q$. This is what I've got so far. First obtain unit vectors $w_1,w_2$ which are orthogonal ...

This problem is about working with smart-phone accelerometers.
To calibrate accelerometer, I need to find three unknown matrices T, K and B that minimize this sum:
$$\sum_{i=0}^N(|g|^2 - |TK(a_i + B)|...

I have 2 images of a scene taken at one moment by two identical cameras (similar cameras intrinsic parameters) by to arbitrary locations and at two arbitrary orientations (different cameras poses).
On ...

Consider a large graph, minimum 1 000 vertices but it can easily go up to 50 000 vertices depending the case. The graph is the result of social relationships (followers, following, friendship) so it ...

I’m trying to prove that the following problem has an integer optimal solution. This will hold if the corresponding linear program would have totally unimodular constraint matrix.
We have $m$ pieces ...

Given a matrix $M \in \mathbb{R}^{n \times m}$ and a set $S \subset \{1, \ldots, n\}$, let $M_{S, {\rm row}}$ be the matrix obtained by picking the rows of $M$ from the set $S$. Similarly, given $S' \...

I have 2 images, called left and right images. I have some matched points $[c_l,r_l]$ and $[c_r,r_r]$ in both of them (these points are in pixel coordinates).
For a 3D point in the real world, they ...

I have been wrestling with this for quite a long time but couldn't convince myself that the following is true:
What I do understand: $\theta_a$ denotes the set of points that are within the ellipsoid....

Say one is given a Vandermonde matrix (https://en.wikipedia.org/wiki/Vandermonde_matrix) of dimension $2^q \times k$ such that the elements of the first column of it are $\{0,1,2,..,-1+2^q\}$. (This ...

I’ve always read advice and warnings about the poor scalability and time spent trying to invert a matrix and why it’s better to solve a system of equations whenever the inverted matrix will be used ...

Is there an efficient algorithm for the following problem?
Given: a $m$-vector $b \in \{0,1,2\}^m$, and a $m \times 2m$ matrix $A$, with the promise that for every $b' \in \{0,1,2\}^m$, there exists $...

I have to solve a system of up to 10000 equations with 10000 unknowns as fast as possible (preferably within a few seconds). I know that Gaussian elimination is too slow for that, so what algorithm is ...